Furnace burner doesn't always come on, so I manually turn the thermostat off/ also turn the temperature setting down, when the furnace shuts off I turn the turn the thermostat back to on/ also turn the temperature setting to the desired room temperature, and then the furnace will usually work properly. What is the problem?

I am going to guess thermostat , fan relay, thermal sensors -- from what you've told me

There is an intermittent condition -- some of the pieces of the puzzle are missing

Brand and model of the thermostat --- brand and model of the furnace

All I can do is guess at the problem -- turn off furnace -- and make a drawing of wires on thermostat

The manufacturer's website may have the info you need --- Use a meter to test thermostat

When you turn the thermostat off and back to on its resetting the control board. You should have a diagnostic light flashing on the control board telling you what's wrong. It's a soft restart if you only have to turn the thermostat off and then back on - check for a dirty flame sensor or dirty burners. If it was major it would not restart from just killing 24 volts on the heating circuit.